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## G = C2×C4⋊1D4order 64 = 26

### Direct product of C2 and C4⋊1D4

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C2×C41D4, C4218C22, C23.8C23, C22.24C24, C24.15C22, C41(C2×D4), (C2×C4)⋊7D4, (C2×C42)⋊11C2, (C22×D4)⋊5C2, (C2×D4)⋊12C22, C22.62(C2×D4), C2.10(C22×D4), (C2×C4).130C23, (C22×C4).123C22, SmallGroup(64,211)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C22 — C2×C4⋊1D4
 Chief series C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C2×C4⋊1D4
 Lower central C1 — C22 — C2×C4⋊1D4
 Upper central C1 — C23 — C2×C4⋊1D4
 Jennings C1 — C22 — C2×C4⋊1D4

Generators and relations for C2×C41D4
G = < a,b,c,d | a2=b4=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >

Subgroups: 441 in 249 conjugacy classes, 105 normal (5 characteristic)
C1, C2, C2, C4, C22, C22, C22, C2×C4, D4, C23, C23, C23, C42, C22×C4, C2×D4, C2×D4, C24, C2×C42, C41D4, C22×D4, C2×C41D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C41D4, C22×D4, C2×C41D4

Character table of C2×C41D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 4K 4L size 1 1 1 1 1 1 1 1 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 -1 1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ3 1 -1 -1 1 1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ4 1 -1 -1 1 1 1 -1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ5 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 -1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ6 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ7 1 1 1 1 1 1 1 1 -1 -1 1 -1 -1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ8 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ10 1 1 1 1 1 1 1 1 1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 -1 linear of order 2 ρ11 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 -1 -1 1 1 -1 -1 -1 -1 -1 -1 1 1 linear of order 2 ρ12 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ13 1 -1 -1 1 1 1 -1 -1 1 -1 -1 -1 1 -1 1 1 -1 1 1 -1 -1 1 -1 1 -1 1 1 -1 linear of order 2 ρ14 1 -1 -1 1 1 1 -1 -1 -1 1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ15 1 -1 -1 1 1 1 -1 -1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ16 1 -1 -1 1 1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 linear of order 2 ρ17 2 2 -2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 -2 2 2 -2 0 0 0 0 orthogonal lifted from D4 ρ18 2 -2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 2 2 -2 -2 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 2 2 0 0 orthogonal lifted from D4 ρ20 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 -2 -2 0 0 0 0 0 0 2 2 orthogonal lifted from D4 ρ21 2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 -2 2 orthogonal lifted from D4 ρ22 2 2 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 0 2 -2 0 0 orthogonal lifted from D4 ρ23 2 -2 2 2 -2 -2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 -2 -2 2 2 0 0 0 0 orthogonal lifted from D4 ρ24 2 2 -2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 2 -2 -2 2 0 0 0 0 orthogonal lifted from D4 ρ25 2 2 -2 -2 2 -2 2 -2 0 0 0 0 0 0 0 0 2 -2 0 0 0 0 0 0 -2 2 0 0 orthogonal lifted from D4 ρ26 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 -2 -2 orthogonal lifted from D4 ρ27 2 -2 2 -2 2 -2 -2 2 0 0 0 0 0 0 0 0 2 2 0 0 0 0 0 0 -2 -2 0 0 orthogonal lifted from D4 ρ28 2 -2 -2 -2 -2 2 2 2 0 0 0 0 0 0 0 0 0 0 -2 2 0 0 0 0 0 0 2 -2 orthogonal lifted from D4

Smallest permutation representation of C2×C41D4
On 32 points
Generators in S32
(1 12)(2 9)(3 10)(4 11)(5 24)(6 21)(7 22)(8 23)(13 18)(14 19)(15 20)(16 17)(25 32)(26 29)(27 30)(28 31)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)
(1 18 31 6)(2 19 32 7)(3 20 29 8)(4 17 30 5)(9 14 25 22)(10 15 26 23)(11 16 27 24)(12 13 28 21)
(1 26)(2 25)(3 28)(4 27)(5 24)(6 23)(7 22)(8 21)(9 32)(10 31)(11 30)(12 29)(13 20)(14 19)(15 18)(16 17)

G:=sub<Sym(32)| (1,12)(2,9)(3,10)(4,11)(5,24)(6,21)(7,22)(8,23)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,18,31,6)(2,19,32,7)(3,20,29,8)(4,17,30,5)(9,14,25,22)(10,15,26,23)(11,16,27,24)(12,13,28,21), (1,26)(2,25)(3,28)(4,27)(5,24)(6,23)(7,22)(8,21)(9,32)(10,31)(11,30)(12,29)(13,20)(14,19)(15,18)(16,17)>;

G:=Group( (1,12)(2,9)(3,10)(4,11)(5,24)(6,21)(7,22)(8,23)(13,18)(14,19)(15,20)(16,17)(25,32)(26,29)(27,30)(28,31), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32), (1,18,31,6)(2,19,32,7)(3,20,29,8)(4,17,30,5)(9,14,25,22)(10,15,26,23)(11,16,27,24)(12,13,28,21), (1,26)(2,25)(3,28)(4,27)(5,24)(6,23)(7,22)(8,21)(9,32)(10,31)(11,30)(12,29)(13,20)(14,19)(15,18)(16,17) );

G=PermutationGroup([[(1,12),(2,9),(3,10),(4,11),(5,24),(6,21),(7,22),(8,23),(13,18),(14,19),(15,20),(16,17),(25,32),(26,29),(27,30),(28,31)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32)], [(1,18,31,6),(2,19,32,7),(3,20,29,8),(4,17,30,5),(9,14,25,22),(10,15,26,23),(11,16,27,24),(12,13,28,21)], [(1,26),(2,25),(3,28),(4,27),(5,24),(6,23),(7,22),(8,21),(9,32),(10,31),(11,30),(12,29),(13,20),(14,19),(15,18),(16,17)]])

Matrix representation of C2×C41D4 in GL5(ℤ)

 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 1 0 0 0 0 0 -1 2 0 0 0 -1 1 0 0 0 0 0 0 1 0 0 0 -1 0
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 -1 0
,
 1 0 0 0 0 0 -1 0 0 0 0 -1 1 0 0 0 0 0 -1 0 0 0 0 0 1

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[1,0,0,0,0,0,-1,-1,0,0,0,2,1,0,0,0,0,0,0,-1,0,0,0,1,0],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,1,0],[1,0,0,0,0,0,-1,-1,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1] >;

C2×C41D4 in GAP, Magma, Sage, TeX

C_2\times C_4\rtimes_1D_4
% in TeX

G:=Group("C2xC4:1D4");
// GroupNames label

G:=SmallGroup(64,211);
// by ID

G=gap.SmallGroup(64,211);
# by ID

G:=PCGroup([6,-2,2,2,2,-2,2,217,103,650,158]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^4=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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